WASHINGTON — The roiling debate over security and liberty did not end with President Barack Obama’s newly announced overhaul of surveillance practices.

Rather, it now enters a volatile next phase as intelligence agencies and a divided Congress try to turn principles into policy.

In responding to months of uproar about government spying, Obama left to be decided the details that would determine just how meaningful the change he promised would be.

He asked security officials to develop ways to protect the privacy of foreigners. He asked Congress to help figure out how to store bulk telephone data. He invited other proposals to restructure a secret intelligence court. All of which means that the future shape of a surveillance apparatus whose secrets have been uncomfortably exposed remains far from certain.

The assurances Obama offered his critics may be made more nebulous by exceptions written into any new policies. The question of what to do with a vast trove of data on everyday Americans may elude policymakers who cannot agree on much. And yet legislators may find their usual politics scrambled by an issue that crosses party lines.

“It’s the beginning of a long process, and the end on some of this is still unclear,” said former Rep. Jane Harman, an author of the last major surveillance law and now the president of the Woodrow Wilson International Center for Scholars. “But the good news is now there’s a full debate in the Congress and in the country about our values and how to address security and liberty at the same time.”

The debate injects the touchy issue into the congressional arena as lawmakers enter an election year. Where once there was little popular demand for reining in the spy agencies, today momentum has built in some quarters after many disclosures by Edward J. Snowden, a former National Security Agency contractor.

Obama’s approach, outlined in a much anticipated speech Friday, leaves intact the spy programs that have stirred so much debate but modifies them in hopes of persuading the public that they will not be abused. The NSA will have to get court permission to search a vast repository of telephone data and will not be able to expand the search as far as it did before.

But Obama had no answer for the biggest question involving the bulk data collection program. Although he said the government should no longer keep the data, he outlined flaws in the only two alternatives floated so far: leaving data with telecommunications providers or creating an independent consortium to store it. He assigned the attorney general, Eric H. Holder Jr., and the director of national intelligence, James R. Clapper Jr., to develop a plan and asked Congress to help.

Unlike many divisive issues to arrive on Capitol Hill, this one appears unlikely to die after waiting for a floor debate that never happens, or be thwarted by the parliamentary maneuver of an uncompromising leader.

“Reformers may not get all of what we want,” said Sen. Richard Blumenthal, D-Conn. “But I think there’s a very real prospect of doing better than the president has proposed, and he’s acknowledged himself that there may be a need for taking additional steps.”

Others were not so certain. “This happens all the time in Washington,” said Sen. Rand Paul, R-Ky. and a vocal critic of surveillance programs. “Everybody gets in an uproar — ‘Congress must act! Congress must act!’ But when they do act, they do something devious and don’t really address the problem.”

Indeed, supporters of the NSA programs say they expect Congress to resist undercutting programs that protect the public. “You will see changes at the margins with significant ambiguities and exceptions that will provide the executive branch with lots of flexibility,” predicted former Rep. Pete Hoekstra, R-Mich., a onetime chairman of the intelligence committee.

What distinguishes the surveillance issue from so many that have stymied a polarized Congress is that it does not follow easy patterns. The libertarian right, represented by Paul, has joined the liberal left, represented by lawmakers like Sen. Bernard Sanders of Vermont, an independent who calls himself a socialist.

On the other side are the leaders of both parties, like Sen. Dianne Feinstein, D-Calif., and Rep. Mike Rogers, R-Mich., leaders of the intelligence committees and supporters of the surveillance programs. The two attended Obama’s speech at the Justice Department, then lingered together afterward consulting with Clapper. Within a few hours, the two issued a joint statement defending the programs.

“It’s interesting because there’s splits within both parties,” said Peter Swire, a Georgia Tech professor who served on a panel of Obama’s that reviewed the surveillance programs. “Potentially it makes it easier because members are open to persuasion. It’s not a party-line vote.”

Sen. Angus King of Maine, an Independent who usually votes with Democrats, said: “Ideology is sort of confusing on this one. When you have Rand Paul and Bernie Sanders on the same side, that makes for a pretty interesting debate.”

Indeed, some Democrats seemed eager to use the issue to distance themselves from an unpopular chief executive. “I don’t believe innocent Alaskans’ personal records need to be collected and analyzed in bulk in an effort to help catch terrorists,” said Sen. Mark Begich of Alaska. “It’s a violation of our civil liberties and is heavy-handed — like using a shark hook to fish for a salmon.”

In the Senate, the debate sets up a possible clash among three of the most powerful and headstrong Democrats: Feinstein; Patrick J. Leahy of Vermont, chairman of the Judiciary Committee and an advocate for expansive reforms; and Harry Reid of Nevada, the majority leader, who feels pinched between the White House and members of his caucus.

In the House, the issue pits Speaker John A. Boehner of Ohio against some in his Republican caucus. When the House debated a bill that would have blocked the NSA from collecting bulk telephone data, the measure came within a dozen votes of passing. Although speakers rarely vote, Boehner took the unusual step of voting in opposition.

Boehner has not endorsed the legislation that seems most likely to be the vehicle for the debate over NSA practices this year, which has been drafted by Leahy and Rep. F. James Sensenbrenner of Wisconsin, a senior Republican who as an author of the Patriot Act holds considerable sway with his colleagues. The bill would end bulk data collection and establish an independent counsel to argue against government requests at the Foreign Intelligence Surveillance Court.

“The bottom line is real reform cannot be done by presidential fiat,” Sensenbrenner said, bluntly making the case that Congress has to act where the White House stopped short. “The president and intelligence community have repeatedly misled Congress and the American people and lack credibility for reform.”

Rep. Adam B. Schiff, D-Calif., who also has sponsored legislation that would curb surveillance programs, said lawmakers may still not go far. “I’m not all that sanguine about Congress’s ability to step up to the plate and enact reform on its own,” he said. “Congress will do some of the easy things,” like require more transparency in surveillance.

But it may not be until next year, when the Patriot Act comes up for renewal, that more significant issues are addressed. “Unequivocal defenders know the program disappears in 18 months,” Schiff said. “That may make them more amenable to compromise.”